Poisson distribution


Story

The number \(n\) of arrivals of a Poisson process in unit time is Poisson distributed.


Example

The number of mutations in a strand of DNA per unit length (since mutations are rare) are Poisson distributed.


Parameters

The single parameter is the rate \(\lambda\) of the arrival of the Poisson process.


Support

The Poisson distribution is supported on the set of nonnegative integers.


Probability mass function

\[\begin{align} f(n;\lambda) = \frac{\lambda^n}{n!}\,\mathrm{e}^{-\lambda}. \end{align}\]

Moments

Mean: \(\lambda\)

Variance: \(\lambda\)


Usage

Package

Syntax

NumPy

rg.poisson(lam)

SciPy

scipy.stats.poisson(lam)

Stan

poisson(lam)



PMF and CDF plots